Subgroups — SL(2,𝔽₃)×#733 #14517219

Order 1944.

211 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×3₋¹⁺²	C₃	1	3	yes	maximal
2	C₃×C₉×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	Q₈×#733	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619581	C₃	1	6	yes	maximal, complemented
5	#10619576	C₃	1	2	yes	maximal, complemented
6	C₂×C₃×#733	—	4	1	no	maximal
7	C₄×#733	—	3	1	no	—
8	C₃×#733	—	4	1	no	Sylow
9	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
10	Q₈×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
12	#6204	C₃²	1	6	yes	—
13	C₃×Q₈⋊C₉	C₃²	1	2	yes	—
14	C₉⋊SL(2,𝔽₃)	—	3	6	no	—
15	C₉×SL(2,𝔽₃)	—	3	2	no	—
16	C₉×SL(2,𝔽₃)	—	3	1	no	—
17	C₉×SL(2,𝔽₃)	—	3	3	no	—
18	C₂×C₃×3₋¹⁺²	—	4	3	no	—
19	C₂×C₃²×C₉	—	4	1	no	—
20	C₂×#733	—	4	6	no	—
21	C₂×#733	—	4	2	no	—
22	C₂×#733	A₄	1	1	yes	characteristic
23	C₄×3₋¹⁺²	—	3	3	no	—
24	C₃×C₄×C₉	—	3	1	no	—
25	C₃×3₋¹⁺²	—	4	3	no	—
26	C₃²×C₉	—	4	1	no	—
27	#733	—	4	6	no	—
28	#733	—	4	2	no	—
29	#733	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
30	C₃²×Q₈	C₃³	1	1	yes	characteristic
31	C₃×SL(2,𝔽₃)	—	3	3	no	—
32	C₃×SL(2,𝔽₃)	H(3,𝔽₃)	1	2	yes	—
33	C₃×SL(2,𝔽₃)	H(3,𝔽₃)	1	1	yes	characteristic
34	Q₈×C₉	—	3	1	no	—
35	Q₈×C₉	—	3	3	no	—
36	Q₈⋊C₉	—	3	2	no	—
37	Q₈⋊C₉	—	3	6	no	—
38	C₂×C₃³	—	4	1	no	—
39	C₂×3₋¹⁺²	—	4	6	no	—
40	C₂×3₋¹⁺²	—	12	6	no	—
41	C₂×3₋¹⁺²	C₃×A₄	1	3	yes	—
42	C₂×C₃×C₉	—	4	2	no	—
43	C₂×C₃×C₉	—	12	1	no	—
44	C₂×C₃×C₉	—	12	2	no	—
45	C₂×C₃×C₉	—	12	3	no	—
46	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
47	C₃²×C₄	—	3	1	no	—
48	C₄×C₉	—	9	1	no	—
49	C₄×C₉	—	9	3	no	—
50	C₃³	—	4	1	no	—
51	3₋¹⁺²	—	4	6	no	—
52	3₋¹⁺²	—	12	6	no	—
53	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	3	yes	—
54	C₃×C₉	—	4	2	no	—
55	C₃×C₉	—	12	1	no	—
56	C₃×C₉	—	12	2	no	—
57	C₃×C₉	—	12	3	no	—
58	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
59	C₃×Q₈	—	3	1	no	—
60	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	SL(2,𝔽₃)	—	3	2	no	—
62	SL(2,𝔽₃)	#733	1	3	yes	direct product factor
63	C₂×C₃²	—	4	1	no	—
64	C₂×C₃²	—	4	2	no	—
65	C₂×C₃²	—	12	3	no	—
66	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
67	C₂×C₉	—	3	1	no	—
68	C₂×C₉	—	3	3	no	—
69	C₂×C₉	—	12	2	no	—
70	C₂×C₉	—	12	6	no	—
71	C₃×C₄	—	3	1	no	—
72	C₃×C₄	—	9	1	no	—
73	C₃²	—	4	1	no	—
74	C₃²	—	4	2	no	—
75	C₃²	—	12	3	no	—
76	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
77	C₉	—	3	1	no	—
78	C₉	—	3	3	no	—
79	C₉	—	12	2	no	—
80	C₉	—	12	6	no	—
81	Q₈	C₃×#733	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	3	1	no	—
83	C₂×C₃	—	4	3	no	—
84	C₂×C₃	—	12	2	no	—
85	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
86	C₄	—	3	1	no	—
87	C₃	—	3	1	no	—
88	C₃	—	4	3	no	—
89	C₃	—	12	2	no	—
90	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
91	C₂	A₄×#733	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	Q₈×#733	C₃
prime radical	C₃²×SL(2,𝔽₃)	C₃²
hypercenter	C₂×#733	A₄
hypocenter	Q₈	C₃×#733
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#733

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. Q₈×#733	C₃	semidirect
4. #10619581	C₃	semidirect
5. #10619576	C₃	semidirect
29. #733	SL(2,𝔽₃)	direct (N × H)
62. SL(2,𝔽₃)	#733	direct (N × H)
81. Q₈	C₃×#733	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
SL(2,𝔽₃)×3₋¹⁺²	3
C₃×C₉×SL(2,𝔽₃)	3
Q₈×#733	3
#10619581	3
#10619576	3
C₂×C₃×#733	4

Subgroups — SL(2,𝔽3)×#733 #14517219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽₃)×#733 #14517219